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Affine Geometry CHAPTER II AFFINE GEOMETRY In the previous chapter we indicated how several basic ideas from geometry have natural interpretations in terms of vector spaces and linear algebra. This chapter continues the process of formulating basic geometric concepts in such terms. It begins with standard material, moves on to consider topics not covered in most courses on classical deductive geometry or analytic geometry, and it concludes by giving an abstract formulation of the concept of geometrical incidence and closely related issues. 1. Synthetic affine geometry In this section we shall consider some properties of Euclidean spaces which only depend upon the axioms of incidence and parallelism Definition. A three-dimensional incidence space is a triple (S; L; P) consisting of a nonempty set S (whose elements are called points) and two nonempty disjoint families of proper subsets of S denoted by L (lines) and P (planes) respectively, which satisfy the following conditions: (I { 1) Every line (element of L) contains at least two points, and every plane (element of P) contains at least three points. (I { 2) If x and y are distinct points of S, then there is a unique line L such that x; y 2 L. Notation. The line given by (I {2) is called xy. (I { 3) If x, y and z are distinct points of S and z 62 xy, then there is a unique plane P such that x; y; z 2 P . (I { 4) If a plane P contains the distinct points x and y, then it also contains the line xy. (I { 5) If P and Q are planes with a nonempty intersection, then P \ Q contains at least two points. Of course, the standard example in R3 with lines and planes defined by the formulas in Chapter I (we shall verify a more general statement later in this chapter). A list of other simple examples appears in Prenowitz and Jordan, Basic Concepts of Geometry, pp. 141{146. A few theorems in Euclidean geometry are true for every three-dimensional incidence space. The proofs of these results provide an easy introduction to the synthetic techniques of these notes. In the first six results, the triple (S; L; P) denotes a fixed three-dimensional incidence space. Definition. A set B of points in S is collinear if there is some line L in S such that B ⊂ L, and it is noncollinear otherwise. A set A of points in S is coplanar if there is some plane P in S such that A ⊂ P , and it is noncoplanar otherwise. | Frequently we say that the points x; y; · · · (etc:) are collinear or coplanar if the set with these elements is collinear or coplanar respectively. 7 8 II. AFFINE GEOMETRY Theorem II.1. Let x, y and z be distinct points of S such that z 62 xy. Then f x; y; z g is a noncollinear set. Proof. Suppose that L is a line containing the given three points. Since x and y are distinct, by (I { 2) we know that L = xy. By our assumption on L it follows that z 2 L; however, this contradicts the hypothesis z 62 xy. Therefore there is no line containing x, y and z. Theorem II.2. There is a subset of four noncoplanar points in S. Proof. Let P be a plane in S. We claim that P contains three noncollinear points. By (I { 1) we know that P contains three distinct points a; b; c0. If these three points are noncollinear, let c = c0. If they are collinear, then the line L containing them is a subset of P by (I { 4), and since L and P are disjoint it follows that L must be a proper subset of P ; therefore there is some point c 2 P such that c 62 L, and by the preceding result the set f a; b; c g is noncollinear. Thus in any case we know that P contains three noncollinear points. Since P is a proper subset of S, there is a point d 2 S such that d 62 P . We claim that f a; b; c d g is noncoplanar. For if Q were a plane containing all four points, then a; b; c 2 P would imply P = Q, which contradicts our basic stipulation that d 62 P . Theorem II.3. The intersection of two distinct lines in S is either a point or the empty set. Proof. Suppose that x 6= y but both belong to L \ M for some lines L and M. By property (I { 2) we must have L = M. Thus the intersection of distinct lines must consist of at most one point. Theorem II.4. The intersection of two distinct planes in S is either a line or the empty set. Proof. Suppose that P and Q are distinct planes in S with a nonempty intersection, and let x 2 P \ Q. By (I { 5) there is a second point y 2 P \ Q. If L is the line xy, then L ⊂ P and L ⊂ Q by two applications of (I { 4); hence we have L ⊂ P \ Q. If there is a point z 2 P \ Q with z 62 L, then the points x, y and z are noncollinear but contained in both of the planes P and Q. By (I { 3) we must have P = Q. On the other hand, by assumption we know P 6= Q, so we have reached a contradiction. The source of this contradiction is our hypothesis that P \ Q strictly contains L, and therefore it follows that P \ Q = L. Theorem II.5. Let L and M be distinct lines, and assume that L \ M 6= ?. Then there is a unique plane P such that L ⊂ P and M ⊂ P . In less formal terms, given two intersecting lines there is a unique plane containing them. Proof. Let x 2 L \ M be the unique common point (it is unique by Theorem 3). By (I { 2) there exist points y 2 L and z 2 M, each of which is distinct from x. The points x, y and z are noncollinear because L = xy and z 2 M − fxg = M − L. By (I { 3) there is a unique plane P such that x; y; z 2 P , and by (I { 4) we know that L ⊂ P and M ⊂ P . This proves the existence of a plane containing both L and M. To see this plane is unique, observe that every plane Q containing both lines must contain x, y and z. By (I { 3) there is a unique such plane, and therefore we must have Q = P . 1. SYNTHETIC AFFINE GEOMETRY 9 Theorem II.6. Given a line L and a point z not on L, there is a unique plane P such that L ⊂ P and z 2 P . Proof. Let x and y be distinct points of L, so that L = xy. We then know that the set f x; y; z g is noncollinear, and hence there is a unique plane P containing them. By (I { 4) we know that L ⊂ P and z 2 P . Conversely, if Q is an arbitrary plane containing L and z, then Q contains the three noncollinear points x, y and z, and hence by (I { 3) we know that Q = P . Notation. We shall denote the unique plane in the preceding result by Lz. Of course, all the theorems above are quite simple; their conclusions are probably very clear intuitively, and their proofs are fairly straightforward arguments. One must add Hilbert's Ax- ioms of Order or the Euclidean Parallelism Axiom to obtain something more substantial. Since our aim is to introduce the parallel postulate at an early point, we might as well do so now (a thorough treatment of geometric theorems derivable from the Axioms of Incidence and Order appears in Chapter 12 of Coxeter, Introduction to Geometry; we shall discuss the Axioms of Order in Section VI.6 of these notes). Definition. Two lines in a three-dimensional incidence space S are parallel if they are disjoint and coplanar (note particularly the second condition). If L and L0 are parallel, we shall write LjjL0 and denote their common plane by LL0. Note that if LjjM then MjjL because the conditions in the definition of parallelism are symmetric in the two lines. Affine three-dimensional incidence spaces Definition. A three-dimensional incidence space (S; L; P) is an affine three-space if the following holds: (EPP) For each line L in S and each point x 62 L there is a unique line L0 ⊂ Lx such that x 2 L and L \ L0 = ? (in other words, there is a unique line L0 which contains x and is parallel to L). This property is often called the Euclidean Parallelism Property, the Euclidean Parallel Postu- late or Playfair's Postulate (see the previously cited online reference for background on these names). A discussion of the origin of the term “affine” appears in Section II.5 of the following online site: http://math.ucr.edu/∼res/math133/geomnotes2b.pdf Many nontrivial results in Euclidean geometry can be proved for arbitrary affine three-spaces. We shall limit ourselves to two examples here and leave others as exercises. In Theorems 7 and 8 below, the triple (S; L; P) will denote an arbitrary affine three-dimensional incidence space. Theorem II.7. Two lines which are parallel to a third line are parallel. 10 II. AFFINE GEOMETRY Proof. There are two cases, depending on whether or not all three lines lie in a single plane; to see that the three lines need not be coplanar in ordinary 3-dimensional coordinate geometry, consider the three lines in R3 given by the z-axis and the lines joining (1; 0; 0) and (0; 1; 0) to (1; 0; 1) and (0; 1; 1) respectively.
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